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This is a companion paper to my preceding one on Harriot's experimentations in the field of the sign-rule of multiplication in algebra. Cardano had earlier attacked the conventional rule in a chapter (to which Harriot refers) of his De Aliza regula liber, published in 1570 as an appendix to the second edition of his Ars magna. He returned to the subject in a brief tract, published nearly a century later in his collected works as Sermo de plus et minus. Only Cardano's valid contention that the standard 'proof of the accustomed sign-rule was faulty made impact, in the Euclidean background to which the 'proof belonged. His cause was to preserve the purity of the Euclidean tradition. It overshadows the true aim in the De Aliza chapter, but the Sermo reveals it plainly as far from retrograde: be wished to retain all the new techniques of algebra that brought in not only 'minus' quantities but also their square roots, and to escape the 'impossible' status of these last. This the rebel sign-rule did for him by making every root of a 'minus' just another 'minus'. [ABSTRACT FROM AUTHOR]

The manuscripts of Harriot discussed in this paper are essentially rough notes marginal to his systematic treatment of algebra, of which a small part was published posthumously The central theme is the sign-rule for multiplication; but the incidentals open up an aspect of symbolism in mathematics entirely new for the time. A more restricted aspect of the same theme was touched on by Commandino in his Euclid, quoted by Harriot as rightly blaming 'those that thinke that minus per minus shal produce plus'; Commandino's authority was Cardano, in a similarly marginal departure from his great Ars magna, the de Aliza, which Harriot also cites in the same reference. The ostensible motivation, to preserve intact the Euclidean tradition in geometry, plays a very minor role in Harriot's notes. The basic aim, to elude the fatal link with imaginaries, they do touch on. More dimly underlying the arguments are the crucial logical facets of the problem, which Harriot faced with a greater conceptual clarity, formulating in words, however, only his own conclusion: the choice of sign-rule is the user's--a fact not fully recognised by mathematicians generally until the 19th century. [ABSTRACT FROM AUTHOR]

Focuses on the works of Nathaniel Torpoley's congestor analyticus and Thomas Harriot's de triangulis laterum rationalium. Manuscripts on numerical problems; Examples of classical problem in number theory; Illustration of the Pythagorean triples presented by Thomas Harriot.